Permutation Representations from Partitions

First, let . This is a pretty trivial “partition”, consisting of one piece of length . The Ferrers diagram of looks like

Any Young tableau thus contains all numbers on the single row, so they’re all row-equivalent. There is only one Young tabloid:

We conclude that is a one-dimensional vector space with the trivial action of .

Next, let — another simple partition with parts of length each. The Ferrers diagram looks like

Now in every Young tableau each number is on a different line, so no two tableaux are row-equivalent. They each give rise to their own Young tabloid, such as

These tabloids correspond to permutations; a generic one looks like

The action of on these tabloids is basically the same as left-multiplication on the underlying set . And so we find the left regular representation.

Finally, consider the partition . This time the Ferrers diagram looks like

and a sample Young tabloid looks like

Any Young tabloid of shape is uniquely specified by the single entry on the second row. Any permutation shuffles them around exactly like it does these entries, and so is isomorphic to the defining representation.

[…] of generalized tableaux of the vector space is in bijection with the basis of -tabloids of the vector space . And this space carries an action of — the linear extension of the action on tabloids. We […]

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